## Relativistic Invariance

 Originally posted by turin But now it does make sense to me that the points in the manifold exist, period. The coordinate system is a way to keep track of them, like assigning an index to the elements of a set?
yeah, this sounds good. the points in the manifold are the points, and there may be many different coordinate systems that you can use to label the point, but we think of the point existing in some sense independently of which coordinate system we use.

 Since the points are not discrete elements, we can't use an index (at least not like i = 1,2,3,...), so we use a continuous variable and call it a coordinate. Just like I could index the elements of a set differently, I can use a different coordinate.
yep

 But now I am confused about why one needs two coordinates instead of one.
you need n coordinates, where n is the dimension of the manifold.

 Is there no way to use just one coordinate to indicate points.
only if the manifold is 1 dimensional (a curve)
 I am having trouble shifting my mind from elements of a set needing only one index and points in a manifold needing more than one coordinate.
if you label the elements of the set A with an index i, then you would use two indices (i,j) to label the elements of the set AxA. in the same sense, you need one real number to specify a point on the line (=R), 2 real numbers to specify a point on the plane (=RxR), etc.

i guess in a set theoretic sense, you don t actually need two indices. the cartesian product of two sets has the same cardinality, so you could count the elements of the new set with the same index, if you had a mind to, but this would muck up the other nice properties of the set, so it is much preferred to just stick to the double index concept.

 (I am at this point going to assume that my analogy was correct that points:manifold::elements:set)
yes, its a good analogy
 In a vector space, I've got the idea of linear independence dictating how many independent vectors I will need in my basis to span the vector space, so the dimension is clear. I don't see the same indicator in a manifold.
perhaps it would be useful to know the definition of a manifold. leaving out some technical details, the definition is this: an n-dimensional manifold is a space X such that for any point x in X, there is a neigborhood of x U and a continous invertible map with continuous inverse (homeomorphism) between U and Rn

in other words, if you look at a manifold up closely enough, it looks like Rn. these local homeomorphisms are the charts of the manifold, they are the coordinates.

 Can you explain the collection of functions a bit?
perhaps it would be useful to see an example. a sphere can be written as the map $(u,v)\mapsto(\sin u\cos v,\sin u\sin v,cos v)$. here u and v are the coordinates. there are two of them because a sphere is a 2 dimensional manifold

 But I thought that the tangent planes for any two points, no matter how close they are, have absolutely nothing to do with each other.
thats right. nothing at all (unless you like chroots approximations above. i don t.)

 Unless I'm allowed to, in some sense, take the tangent plane with me, then I have to go from one to another. But, if adjacent tangent planes have nothing to do with each other, then this process seems disjoint, abrupt, or something. There just seems to be a discontinuity here, but I can't put my finger on it. I'll have to let this idea soak in my mind, too.
i m not sure what exactly you re saying here. i can assure you that, for example, if you have a smooth curve on the manifold, then this curve will have a tangent vector which is in the tangent space to the manifold at each point the curve passes through. furthermore, this tangent vector to the curve will vary from tangent space to tangent space, as the curve goes along, in a smooth way.

there is no discontinuity. perhaps it would be useful to know the following fact: a smooth manifold, together with the tangent space at each point, considered as one larger set, is also a smooth manifold. this manifold is called the tangent bundle.

 Originally posted by turin What vector fields? I only know of vector fields like the electric field and such. Can you explain these vector fields?
a vector field is any mapping that assigns a vector to each point of the space. electric field is an example of a vector field on minkowski space, sure. you know how every vector space has a basis? a nice basis for cartesian space is i, j, and k. these are the unit vectors pointing along each coordinate. since there is a tangent space to every point in space, these form a basis at every point in space. hence they are vector fields (not just vectors). the vectors Ambitwistor are the same thing for polar coordinates.

this should be apparent to you. if you know that electric field is a vector field, and you know that electric field can be written $\mathbf{E}=E_x\hat{\imath}+E_y\hat{\jmath}+E_z\hat{k}$, so those i, j, and k are also vector fields. you can choose to work in polar coordinates, and thats all we are doing here.

 Doesn't the fact that you're trying to build coordinates by using vectors automatically tell you that it doesn't make sense? I thought that was the lesson I was learning on the sphere.
points in the sphere are not vectors. never the less, given points in the sphere, you can calculate the tangent vectors, and given tangent vectors, you can calculate coordinates for points in the sphere.

 I don't understand commuting/noncommuting vectors. The only experience I have with nontrivial commutation is in QM, but I only understand the concept with operators, not vectors. Can you explain?
vectors are differential operators. if you would like to understand why, well, i explained this concept at length once, when i started writing a tutorial for differential forms, here.

in short, there are various things you can do to define tangent vectors in an intrinsic way. a nice starting way is to define a tangent vector as an equivalence class of curves all of which have the same derivative at a point. with a little work, you see that this is canonically isomorphic to the space of first order differential operators on functions on the manifold. therefore i can simply say they are the same thing, and call a tangent vector on a manifold a differential operator.

this is tremendously useful, though unfamiliar.